1Department of Computational Methods, Institute of Mathematics, National University of Uzbekistan, Tashkent, Uzbekistan
American Journal of Numerical Analysis.
2013,
Vol. 1 No. 1, 22-31
DOI: 10.12691/ajna-1-1-4
Copyright © 2013 Science and Education PublishingCite this paper: Kholmat M. Shadimetov, Abdullo R. Hayotov, Dilshod M. Akhmedov. Optimal Quadrature Formulas for the Cauchy Type Singular Integral in the Sobolev Space
L2(2)(-1,1).
American Journal of Numerical Analysis. 2013; 1(1):22-31. doi: 10.12691/ajna-1-1-4.
Correspondence to: Abdullo R. Hayotov, Department of Computational Methods, Institute of Mathematics, National University of Uzbekistan, Tashkent, Uzbekistan. Email:
hayotov@mail.ruAbstract
This paper studies the problem of construction of the optimal quadrature formula in the sense of Sard in L2(2)(-1,1) S.L.Sobolev space for approximate calculation of the Cauchy type singular integral. Using the discrete analogue of the operator d4/dx4 we obtain new optimal quadrature formulas. Furthermore, explicit formulas of the optimal coefficients are obtained. Finally, in numerical examples, we give the error bounds obtained for the case h=0.02 by our optimal quadrature formula and compared with the corresponding error bounds of the quadrature formula (15) of the work [26] at different values of singular point t. The numerical results show that our quadrature formula is more accurate than the quadrature formula constructed in the work [26].
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